Yes I do think I need to be more assertive when I pick her up and not to just let her refuse to come.An idea:
Bring the star chart with you and get her to put stars on for things she already did that were good. Maybe ask her what other stars she wants to get today.
If things do not go smoothly, isn't there a way you could put your foot down. "Mummy and Daddy have already arranged what is happening for the next few days so we need to go now. You are a big clever, girl and it will all be ok. We cannot change the arrangement."
I think some way must be found of overriding your ex's transfer of her battle with you onto the child.
I have a few ideas of what to do today and things to say which will be respectful as always but less defeatist.